As Europe grounded most airline flights for a fourth day Sunday because of a volcanic ash cloud spreading from Iceland, increasingly desperate airlines ran test flights to show that flying was safe and pressed aviation authorities to loosen the flight ban.

Airlines complained that European governments were overreacting, relying on incomplete data from computer models rather than real-world safety tests in the air above Europe. In a blunt statement Sunday, representatives of Europe’s airlines and airports called for “an immediate reassessment of the present restrictions.”